One-Dimensional Schrödinger Operators with Complex Potentials

We discuss realizations of $$L:=-\partial _x^2+V(x)$$ as closed operators on $$L^2]a,b[$$, where V is complex, locally integrable and may have an arbitrary behavior at (finite or infinite) endpoints a and b. The main tool of our analysis is Green’s operators, that is, various right inverses of L.